5 - Other Aspects of Simple Modules

Summary

So far we have focused attention mainly on the classification and description of simple modules for a group of Lie type. This chapter assembles a number of more-or-less independent topics which emerge naturally from this study:

• In 5.1 we look at how Frobenius maps permute simple G-modules of arbitrary highest weight. This contributes in turn to a determination of minimal splitting fields (5.2).

• When there are two root lengths, with p equal to the squared ratio of long to short, special isogenies exist between algebraic groups with corresponding dual root systems. This leads to a refined tensor product factorization of their simple modules. For the nonisomorphic groups of types B_ℓ and C_ℓ with ℓ > 2 in characteristic 2, we then find a dimension-preserving bijection between the two collections of simple modules (5.3–5.4).

• After recalling some standard facts about Grothendieck rings and Brauer characters (5.5), we compare formal characters for G-modules with Brauer characters for G^F (5.6).

• Several sections explore a recurring theme: the behavior of G-modules on restriction to a finite group of Lie type. Examples include simple modules L(λ) for arbitrary λ ∈ X⁺, tensor products of simple modules, Weyl modules. Further instances will arise later in the study of projective modules, Ext groups, etc.

When dealing with an arbitrary KG-module M we use the notation [M : L]_G for the number of composition factors of M isomorphic to a given simple module L, and similarly with G replaced by the algebraic group G.